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Mathematical models are essential for understanding and making predictions about systems arising in nature and engineering. Yet, mathematical models are a simplification of true phenomena, thus making predictions subject to uncertainty. Hence, the ability to quantify uncertainties is essential to any modelling framework, enabling the user to assess the importance of certain parameters on quantities of interest and have control over the quality of the model output by providing a rigorous understanding of uncertainty. Peridynamic models are a particular class of mathematical models that have proven to be remarkably accurate and robust for a large class of material failure problems. However, the high computational expense of peridynamic models remains a major limitation, hindering outer-loop applications that require a large number of simulations, for example, uncertainty quantification. This contribution provides a framework to make such computations feasible. By employing a Multilevel Monte Carlo (MLMC) framework, where the majority of simulations are performed using a coarse mesh, and performing relatively few simulations using a fine mesh, a significant reduction in computational cost can be realised, and statistics of structural failure can be estimated. The results show a speed-up factor of 16x over a standard Monte Carlo estimator, enabling the forward propagation of uncertain parameters in a computationally expensive peridynamic model. Furthermore, the multilevel method provides an estimate of both the discretisation error and sampling error, thus improving the confidence in numerical predictions. The performance of the approach is demonstrated through an examination of the statistical size effect in quasi-brittle materials.

Hyperuniformity is the study of stationary point processes with a sub-Poisson variance in a large window. In other words, counting the points of a hyperuniform point process that fall in a given large region yields a small-variance Monte Carlo estimation of the volume. Hyperuniform point processes have received a lot of attention in statistical physics, both for the investigation of natural organized structures and the synthesis of materials. Unfortunately, rigorously proving that a point process is hyperuniform is usually difficult. A common practice in statistical physics and chemistry is to use a few samples to estimate a spectral measure called the structure factor. Its decay around zero provides a diagnostic of hyperuniformity. Different applied fields use however different estimators, and important algorithmic choices proceed from each field's lore. This paper provides a systematic survey and derivation of known or otherwise natural estimators of the structure factor. We also leverage the consistency of these estimators to contribute the first asymptotically valid statistical test of hyperuniformity. We benchmark all estimators and hyperuniformity diagnostics on a set of examples. In an effort to make investigations of the structure factor and hyperuniformity systematic and reproducible, we further provide the Python toolbox structure_factor, containing all the estimators and tools that we discuss.

We analyze the complexity of sampling from a class of heavy-tailed distributions by discretizing a natural class of It\^o diffusions associated with weighted Poincar\'e inequalities. Based on a mean-square analysis, we establish the iteration complexity for obtaining a sample whose distribution is $\epsilon$ close to the target distribution in the Wasserstein-2 metric. In this paper, our results take the mean-square analysis to its limits, i.e., we invariably only require that the target density has finite variance, the minimal requirement for a mean-square analysis. To obtain explicit estimates, we compute upper bounds on certain moments associated with heavy-tailed targets under various assumptions. We also provide similar iteration complexity results for the case where only function evaluations of the unnormalized target density are available by estimating the gradients using a Gaussian smoothing technique. We provide illustrative examples based on the multivariate $t$-distribution.

The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. We propose a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from each contaminated distribution. Under standard moment assumptions, $\mathsf{W}_p^\varepsilon$ is shown to achieve strong robust estimation guarantees under the Huber $\varepsilon$-contamination model. Our formulation of this robust distance amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of $\mathsf{W}_p^\varepsilon$, encompassing robustness guarantees, characterization of optimal perturbations, regularity, duality, and statistical estimation. In particular, by decoupling the optimization variables, we arrive at a simple dual form for $\mathsf{W}_p^\varepsilon$ that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.

In this article, we construct semiparametrically efficient estimators of linear functionals of a probability measure in the presence of side information using an easy empirical likelihood approach. We use estimated constraint functions and allow the number of constraints to grow with the sample size. Considered are three cases of information which can be characterized by infinitely many constraints: (1) the marginal distributions are known, (2) the marginals are unknown but identical, and (3) distributional symmetry. An improved spatial depth function is defined and its asymptotic properties are studied. Simulation results on efficiency gain are reported.

Mean field control (MFC) problems have been introduced to study social optima in very large populations of strategic agents. The main idea is to consider an infinite population and to simplify the analysis by using a mean field approximation. These problems can also be viewed as optimal control problems for McKean-Vlasov dynamics. They have found applications in a wide range of fields, from economics and finance to social sciences and engineering. Usually, the goal for the agents is to minimize a total cost which consists in the integral of a running cost plus a terminal cost. In this work, we consider MFC problems in which there is no terminal cost but, instead, the terminal distribution is prescribed. We call such problems mean field optimal transport problems since they can be viewed as a generalization of classical optimal transport problems when mean field interactions occur in the dynamics or the running cost function. We propose three numerical methods based on neural networks. The first one is based on directly learning an optimal control. The second one amounts to solve a forward-backward PDE system characterizing the solution. The third one relies on a primal-dual approach. We illustrate these methods with numerical experiments conducted on two families of examples.

In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order $1/T$ up to logarithmic factors on the objective function scale, where $T$ denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository \texttt{LogConcComp} (Log-Concave Computation).

We adopt an information-theoretic framework to analyze the generalization behavior of the class of iterative, noisy learning algorithms. This class is particularly suitable for study under information-theoretic metrics as the algorithms are inherently randomized, and it includes commonly used algorithms such as Stochastic Gradient Langevin Dynamics (SGLD). Herein, we use the maximal leakage (equivalently, the Sibson mutual information of order infinity) metric, as it is simple to analyze, and it implies both bounds on the probability of having a large generalization error and on its expected value. We show that, if the update function (e.g., gradient) is bounded in $L_2$-norm, then adding isotropic Gaussian noise leads to optimal generalization bounds: indeed, the input and output of the learning algorithm in this case are asymptotically statistically independent. Furthermore, we demonstrate how the assumptions on the update function affect the optimal (in the sense of minimizing the induced maximal leakage) choice of the noise. Finally, we compute explicit tight upper bounds on the induced maximal leakage for several scenarios of interest.

With apparently all research on estimation-of-distribution algorithms (EDAs) concentrated on pseudo-Boolean optimization and permutation problems, we undertake the first steps towards using EDAs for problems in which the decision variables can take more than two values, but which are not permutation problems. To this aim, we propose a natural way to extend the known univariate EDAs to such variables. Different from a naive reduction to the binary case, it avoids additional constraints. Since understanding genetic drift is crucial for an optimal parameter choice, we extend the known quantitative analysis of genetic drift to EDAs for multi-valued variables. Roughly speaking, when the variables take $r$ different values, the time for genetic drift to become significant is $r$ times shorter than in the binary case. Consequently, the update strength of the probabilistic model has to be chosen $r$ times lower now. To investigate how desired model updates take place in this framework, we undertake a mathematical runtime analysis on the $r$-valued LeadingOnes problem. We prove that with the right parameters, the multi-valued UMDA solves this problem efficiently in $O(r\log(r)^2 n^2 \log(n))$ function evaluations. Overall, our work shows that EDAs can be adjusted to multi-valued problems, and it gives advice on how to set the main parameters.

Thompson sampling (TS) for the parametric stochastic multi-armed bandits has been well studied under the one-dimensional parametric models. It is often reported that TS is fairly insensitive to the choice of the prior when it comes to regret bounds. However, this property is not necessarily true when multiparameter models are considered, e.g., a Gaussian model with unknown mean and variance parameters. In this paper, we first extend the regret analysis of TS to the model of uniform distributions with unknown supports. Specifically, we show that a switch of noninformative priors drastically affects the regret in expectation. Through our analysis, the uniform prior is proven to be the optimal choice in terms of the expected regret, while the reference prior and the Jeffreys prior are found to be suboptimal, which is consistent with previous findings in the model of Gaussian distributions. However, the uniform prior is specific to the parameterization of the distributions, meaning that if an agent considers different parameterizations of the same model, the agent with the uniform prior might not always achieve the optimal performance. In light of this limitation, we propose a slightly modified TS-based policy, called TS with Truncation (TS-T), which can achieve the asymptotic optimality for the Gaussian distributions and the uniform distributions by using the reference prior and the Jeffreys prior that are invariant under one-to-one reparameterizations. The pre-processig of the posterior distribution is the key to TS-T, where we add an adaptive truncation procedure on the parameter space of the posterior distributions. Simulation results support our analysis, where TS-T shows the best performance in a finite-time horizon compared to other known optimal policies, while TS with the invariant priors performs poorly.